Vector field flow over upper surface of sphere

In summary, the problem is to calculate the flow over the upper surface of a sphere with given vector field and normal vector. Using the Gaussian law and converting to spherical coordinates, the integral on the right side is solved and the integral on the left side is evaluated but appears to have mismatched units. However, this is not a problem as the parameter r is dimensionless and the final result will be a number.
  • #1
skrat
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Homework Statement


Calculate the flow over the upper surface of sphere ##x^2+y^2+z^2=1## with normal vector pointed away from origin. Vector field is given as ##\vec{F}=(z^2x,\frac{1}{3}y^3+tan(z),x^2z+y^2)##

Homework Equations


Gaussian law: ##\int \int _{\partial \Sigma }\vec{F}d\vec{S}=\int \int \int _{\Sigma }\nabla \vec{F}dV##

The Attempt at a Solution



The idea here is to see that ##\int \int _{\partial \Sigma }\vec{F}d\vec{S}+\int\int_O \vec{F}d\vec{S}=\int \int \int _{\Sigma }\nabla \vec{F}dV##

Where I marked the circle (plane) at ##z=0##.

Well, I know how to solve this problem yet I have some problems, let me show you where... Let's start with the integral on the right side. We will need spherical coordinates ##x=rsin\theta cos\phi ##, ##y=rsin\theta sin\phi ## and ##z=rcos\phi ##. The integral is than:

##\int \int \int _{\Sigma }\nabla \vec{F}dV)=\int_{0}^{2\pi }d\phi \int_{0}^{\pi /2}d\theta\int_{0}^{1}(r^2sin^2\theta cos^2\phi +r^2sin^2\theta sin^2\phi +r^2cos\theta )r^2sin\theta dr##

one ##r^2## comes from Jacobian determinant.

Let's forget about all the ##\theta ## and ##\phi ##. Integration by r is obviously ##\frac{r^5}{5}##.

OK.

Now the left side of equation ##\int \int _{\partial \Sigma }\vec{F}d\vec{S}+\int\int_O \vec{F}d\vec{S}=\int \int \int _{\Sigma }\nabla \vec{F}dV##, specifically integral over surface ##O##.

Using polar coordinates ##x=rcos\phi ##, ##y=rsin\phi ## and obviously ##z=0##. The parameterization is than ##r(r,\phi )=(rcos\phi ,rsin\phi , 0)## and ##r_r \times r_{\phi}=(0,0,r)##. Therefore the integral over ##O## for ##z=0##:

##\int\int_O \vec{F}d\vec{S}=-\int_{0}^{2\pi }d\phi \int_{0}^{1}(0,something,r^2sin^2 \phi)(0,0,r)dr##

Jacobian is included in normal vector. Sorry, for using "something" instead of calculating it.

Now let's again forget about all the angles and integrate by ##r##, the integral is than ##\frac{r^4}{4}##..

My question here: WHAT IS WRONG? The units don't match!
 
Last edited:
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  • #2
true, there is an ##r^5## term on the right hand side and a ##r^4## term on the left hand side. But this is not a problem. In this situation, ##r## does not have the dimensions of length. It is dimensionless. Look at the original formula given: ##x^2+y^2+z^2=1## This tells you all you need to know about dimensions. One other thing to keep in mind, is that yes, there is an ##r^4## term on the left hand side, but once you finish the integration, you will use the limit (which is just a number). So you just get numbers on the left hand side and right hand side. No problem.
 

Related to Vector field flow over upper surface of sphere

What is a vector field flow?

A vector field flow is a mathematical concept that describes the movement and direction of a quantity, such as velocity or force, in a specific region of space.

What is the upper surface of a sphere?

The upper surface of a sphere refers to the curved, outermost part of a sphere that is facing away from the center.

How is the vector field flow over the upper surface of a sphere visualized?

The vector field flow over the upper surface of a sphere is typically visualized using arrows or lines to represent the direction and magnitude of the vector field at various points on the surface.

What factors affect the vector field flow over the upper surface of a sphere?

The vector field flow over the upper surface of a sphere can be affected by various factors such as the shape and size of the sphere, the velocity of the flow, and any external forces acting on the sphere.

How is the vector field flow over the upper surface of a sphere used in scientific research?

The vector field flow over the upper surface of a sphere is commonly used in fluid dynamics research to understand the behavior of fluids, such as air or water, as they flow over objects. It can also be used in other areas of physics and engineering to study the effects of forces on spherical objects.

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